You just need a second parallel ATA (aka IDE) drive (not an SATA drive). The link you included says that model of computer you have was introduced in "30-Jan-2003" so it's really not that old. That means that disk capacity should not be an issue. (Some older systems can't handle drives with capacities larger than 134 GB for example.)
A typical motherboard will have two IDE channels (0 and 1). Each IDE channel can support two devices - one master and one slave. That implies a maximum of four IDE devices on a typical system. Some systems have more IDE channels or some IDE and SATA channels. IDE-0 (or the Primary IDE) will be the one that your old hard drive is attached to. Normally, although the cable supports a second device, there won't be anything attached to the second connector. IDE-1 (or the Secondary IDE) will be what any CD-ROM and/or DVD-ROM (or CD-RW or DVD-RW) devices will be attached to.
The above sounds like what you have. Some companies save a few pennies by using an an IDE data cable with only one connector that goes onto the motherboard and only a single device connector rather than two [e.g., Compaq did this]. If that were the case, you'd need to replace the cable with one that has (one motherboard and) two device connectors. (Retail drives will include this cable anyway.)
On your existing disk drive, there is a jumper on the back (near where the drive power and data cables that is used to set whether the drive is the master, slave or cable select (or possibly no jumper at all which will default to master or cable select depending on the drive). Suffice it to say, your existing drive is acting as the master. After you add the new drive, it will become the slave and your new drive will be the master.
The trick is that you need to transfer the OS and all your existing files from your old drive to your new drive if you want to boot from it. Most retail drives will include a utility for doing just that. The one that comes with Seagate drives was very nice and the instructions for migrating from your old drive to your new one were pretty clear. (As I recall, though, it only handled the C: Drive. If you have multiple partitions on your old disk, you'd have have to copy those yourself and move the drive letters around afterwards. It may be improved now as I used it over a year ago.) There are also commercial packages that do this as well.
Case: Enermax CS-A106USB
P/S: OCZ OCZ700GXSSLI 700W
Motherboard: ASUS P5N32-E SLI
CPU: Intel E6300(idle 35C/load 39C)
RAM: OCZ Platinum Rev 2 DDR2 800 MHz (4 x 1GB) 4-4-4-15-T1 (2.1v)
Hard Disk: Seagate 2x250GB SATA ST3250620AS in RAID1
Monitor: Mitsubishi DP900u
Graphics: BFG GeForce 8800GT PCI-E
Sound: SoundBlaster X-Fi Fatal1ty
Speakers: Klipsch ProMedia v.2-400 4.1 speakers
Optical: LiteOn 16X DVD-ROM; Sony DRU-710A DVD-RW